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Contents
Preface xi
1 Introduction 1
1.1 Potential of Nanoscale Engineering . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation for Multiple Scale Modeling . . . . . . . . . . . . . . . . . . . . 2
1.3 Educational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Classical Molecular Dynamics 7
2.1 Mechanics of a System of Particles . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Mechanical Forces and Potential Energy . . . . . . . . . . . . . . . 8
2.1.3 Lagrange Equations ofMotion . . . . . . . . . . . . . . . . . . . . 10
2.1.4 Integrals of Motion and Symmetric Fields . . . . . . . . . . . . . . 12
2.1.5 Newtonian Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Molecular Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Pair-Wise Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Multibody Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Molecular Dynamics Applications . . . . . . . . . . . . . . . . . . . . . . . 28
3 Lattice Mechanics 37
3.1 Elements of Lattice Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Basic Symmetry Principles . . . . . . . . . . . . . . . . . . . . . . 40
3.1.3 Crystallographic Directions and Planes . . . . . . . . . . . . . . . . 42
3.2 Equation of Motion of a Regular Lattice . . . . . . . . . . . . . . . . . . . 42
3.2.1 Unit Cell and the Associate Substructure . . . . . . . . . . . . . . . 43
3.2.2 Lattice Lagrangian and Equations of Motion . . . . . . . . . . . . . 45
3.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.2 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 53
3.4 StandingWaves in Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 NormalModes and Dispersion Branches . . . . . . . . . . . . . . . 55
3.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Green’s Function Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.5.1 Solution for a Unit Pulse . . . . . . . . . . . . . . . . . . . . . . . 59
3.5.2 Free Lattice with Initial Perturbations . . . . . . . . . . . . . . . . . 61
3.5.3 Solution for Arbitrary Dynamic Loads . . . . . . . . . . . . . . . . 61
3.5.4 General Inhomogeneous Solution . . . . . . . . . . . . . . . . . . . 62
3.5.5 Boundary Value Problems and the Time History Kernel . . . . . . . 62
3.5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Quasi-Static Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6.1 Equilibrium State Equation . . . . . . . . . . . . . . . . . . . . . . 66
3.6.2 Quasi-Static Green’s Function . . . . . . . . . . . . . . . . . . . . . 67
3.6.3 Multiscale Boundary Conditions . . . . . . . . . . . . . . . . . . . . 67
4 Methods of Thermodynamics and Statistical Mechanics 79
4.1 Basic Results of the Thermodynamic Method . . . . . . . . . . . . . . . . . 80
4.1.1 State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1.2 Energy Conservation Principle . . . . . . . . . . . . . . . . . . . . . 84
4.1.3 Entropy and the Second Law of Thermodynamics . . . . . . . . . . 86
4.1.4 Nernst’s Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1.5 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Statistics of Multiparticle Systems in Thermodynamic Equilibrium . . . . . 91
4.2.1 Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.2 Statistical Description of Multiparticle Systems . . . . . . . . . . . 93
4.2.3 Microcanonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.4 Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.5 Maxwell–Boltzmann Distribution . . . . . . . . . . . . . . . . . . . 104
4.2.6 Thermal Properties of Periodic Lattices . . . . . . . . . . . . . . . . 107
4.3 Numerical Heat Bath Techniques . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Berendsen Thermostat . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.2 Nos′e–Hoover Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3.3 Phonon Method for Solid–Solid Interfaces . . . . . . . . . . . . . . 119
5 Introduction to Multiple Scale Modeling 123
5.1 MAAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Coarse-GrainedMolecular Dynamics . . . . . . . . . . . . . . . . . . . . . 126
5.3 Quasi-Continuum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 CADD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.5 Bridging Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Introduction to Bridging Scale 131
6.1 Bridging Scale Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1.1 Multiscale Equations of Motion . . . . . . . . . . . . . . . . . . . . 133
6.2 Removing Fine Scale Degrees of Freedom in Coarse Scale Region . . . . . 136
6.2.1 Relationship of Lattice Mechanics to Finite Elements . . . . . . . . 137
6.2.2 LinearizedMD Equation ofMotion . . . . . . . . . . . . . . . . . . 139
6.2.3 Elimination of Fine Scale Degrees of Freedom . . . . . . . . . . . . 141
6.2.4 Commentary on Reduced Multiscale Formulation . . . . . . . . . . 143
6.2.5 Elimination of Fine Scale Degrees of Freedom:
3D Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2.6 Numerical Implementation of Impedance Force . . . . . . . . . . . 150
6.2.7 Numerical Implementation of Coupling Force . . . . . . . . . . . . 151
6.3 Discussion on the Damping Kernel Technique . . . . . . . . . . . . . . . . 152
6.3.1 Programming Algorithm for Time History Kernel . . . . . . . . . . 157
6.4 Cauchy–Born Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.5 Virtual Atom ClusterMethod . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.5.1 Motivations and General Formulation . . . . . . . . . . . . . . . . . 159
6.5.2 General Idea of the VACModel . . . . . . . . . . . . . . . . . . . . 163
6.5.3 Three-Way Concurrent Coupling with QM Method . . . . . . . . . 164
6.5.4 Tight-Binding Method for Carbon Systems . . . . . . . . . . . . . . 167
6.5.5 Coupling with the VACModel . . . . . . . . . . . . . . . . . . . . 169
6.6 Staggered Time Integration Algorithm . . . . . . . . . . . . . . . . . . . . . 170
6.6.1 MD Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.6.2 FE Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Summary of Bridging Scale Equations . . . . . . . . . . . . . . . . . . . . 172
6.8 Discussion on the Bridging ScaleMethod . . . . . . . . . . . . . . . . . . . 173
7 Bridging Scale Numerical Examples 175
7.1 Comments on Time History Kernel . . . . . . . . . . . . . . . . . . . . . . 175
7.2 1D Bridging Scale Numerical Examples . . . . . . . . . . . . . . . . . . . . 176
7.2.1 Lennard-Jones Numerical Examples . . . . . . . . . . . . . . . . . . 176
7.2.2 Comparison of VAC Method and Cauchy–Born Rule . . . . . . . . 178
7.2.3 Truncation of Time History Kernel . . . . . . . . . . . . . . . . . . 179
7.3 2D/3D Bridging Scale Numerical Examples . . . . . . . . . . . . . . . . . . 182
7.4 Two-DimensionalWave Propagation . . . . . . . . . . . . . . . . . . . . . 184
7.5 Dynamic Crack Propagation in Two Dimensions . . . . . . . . . . . . . . . 187
7.6 Dynamic Crack Propagation in Three Dimensions . . . . . . . . . . . . . . 195
7.7 Virtual Atom Cluster Numerical Examples . . . . . . . . . . . . . . . . . . 200
7.7.1 Bending of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . 200
7.7.2 VAC Coupling with Tight Binding . . . . . . . . . . . . . . . . . . 200
8 Non-Nearest Neighbor MD Boundary Condition 203
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2 Theoretical Formulation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.2.1 Force Boundary Condition: 1D Illustration . . . . . . . . . . . . . . 207
8.2.2 Displacement Boundary Condition: 1D Illustration . . . . . . . . . . 210
8.2.3 Comparison to Nearest Neighbors Formulation . . . . . . . . . . . . 211
8.2.4 Advantages of Displacement Formulation . . . . . . . . . . . . . . . 212
8.3 Numerical Examples: 1D Wave Propagation . . . . . . . . . . . . . . . . . 212
8.4 Time-History Kernels for FCC Gold . . . . . . . . . . . . . . . . . . . . . . 213
8.5 Conclusion for the Bridging Scale Method . . . . . . . . . . . . . . . . . . 215
8.5.1 Bridging Scale Perspectives . . . . . . . . . . . . . . . . . . . . . . 220
9 Multiscale Methods for Material Design 223
9.1 Multiresolution Continuum Analysis . . . . . . . . . . . . . . . . . . . . . . 225
9.1.1 Generalized Stress and Deformation Measures . . . . . . . . . . . . 227
9.1.2 Interaction between Scales . . . . . . . . . . . . . . . . . . . . . . . 231
9.1.3 Multiscale Materials Modeling . . . . . . . . . . . . . . . . . . . . 232
9.2 Multiscale Constitutive Modeling of Steels . . . . . . . . . . . . . . . . . . 234
9.2.1 Methodology and Approach . . . . . . . . . . . . . . . . . . . . . . 235
9.2.2 First-Principles Calculation . . . . . . . . . . . . . . . . . . . . . . 235
9.2.3 Hierarchical Unit Cell and Constitutive Model . . . . . . . . . . . . 237
9.2.4 Laboratory Specimen Scale: Simulation and Results . . . . . . . . . 239
9.3 Bio-InspiredMaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.3.1 Mechanisms of Self-Healing inMaterials . . . . . . . . . . . . . . . 244
9.3.2 Shape-Memory Composites . . . . . . . . . . . . . . . . . . . . . . 246
9.3.3 Multiscale Continuum Modeling of SMA Composites . . . . . . . . 250
9.3.4 Issues ofModeling and Simulation . . . . . . . . . . . . . . . . . . 256
9.4 Summary and Future Research Directions . . . . . . . . . . . . . . . . . . . 260
10 Bio–Nano Interface 263
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
10.2 Immersed Finite ElementMethod . . . . . . . . . . . . . . . . . . . . . . . 265
10.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.2.2 Computational Algorithm of IFEM . . . . . . . . . . . . . . . . . . 268
10.3 Vascular Flow and Blood Rheology . . . . . . . . . . . . . . . . . . . . . . 269
10.3.1 HeartModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
10.3.2 Flexible Valve–Viscous Fluid Interaction . . . . . . . . . . . . . . . 270
10.3.3 Angioplasty Stent . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
10.3.4 Monocyte Deposition . . . . . . . . . . . . . . . . . . . . . . . . . 272
10.3.5 Platelet Adhesion and Blood Clotting . . . . . . . . . . . . . . . . . 272
10.3.6 RBC Aggregation and Interaction . . . . . . . . . . . . . . . . . . . 274
10.4 Electrohydrodynamic Coupling . . . . . . . . . . . . . . . . . . . . . . . . 280
10.4.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
10.4.2 Electro-manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10.4.3 Rotation of CNTs Induced by Electroosmotic Flow . . . . . . . . . 285
10.5 CNT/DNA Assembly Simulation . . . . . . . . . . . . . . . . . . . . . . . 287
10.6 CellMigration and Cell–Substrate Adhesion . . . . . . . . . . . . . . . . . 290
10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Appendix A Kernel Matrices for EAM Potential 297
Bibliography 301
Index 315

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Preface
Within the past decade, the emphasis of scientific research worldwide has shifted to the
study of the behavior of materials at the atomic scale of matter. The proliferation of scientists
and engineers studying matter at this length scale has led to the coining of the phrase
nanotechnology. This term can generally be taken to imply the investigation and technological
utilization of the properties of matter at length scales of one thousand nanometers or
smaller. Generally, a few thousand atoms will exist in the space of thousand nanometers.
As engineers typically study the mechanical properties of materials, the corresponding
emphasis of research in the engineering community has been on nano mechanics. The
term “nano mechanics” is typically associated with the study and characterization of the
mechanical behavior of individual atoms, atomic-scale systems and structures in response
to various types of forces and loading conditions. The specific nature of nano mechanics
research generally varies depending on the discipline of the engineer; the topic of interest
can involve the atomic-scale effect of fracture and wear on material performance, mechanical
properties of nanocomposites, atomic-scale flow and locomotion of individual biological
cells.
Regardless of the interest of the particular scientist or engineer, what is universally
agreed upon is the overall potential that nanotechnology, and particularly nano mechanics,
has for the betterment of our society, including the sectors of private industry, national
defense and homeland security. An emphasis on nanoscale entities will make our manufacturing
technologies and infrastructure more sustainable in terms of reduced energy usage
and environmental pollution. Recent advances made by the research community in this
topic have stimulated ever-broader research activities in science and engineering that are
devoted to their development and applications.
Many areas of research are rapidly advancing owing to the combined efforts of science
and engineering. In mechanics and materials, we are particularly excited with the progress
in research and education that can be achieved by combining engineering and basic sciences
through modeling and simulation together with experimentation. Owing to the combination
of constantly increasing computational power and the increased knowledge and understanding
of material behavior, multiple scale modeling methods have recently emerged as the
tool of choice to link the mechanical behavior of materials from the smallest scale of atoms
to the largest scale of structures. Multiple scale methods offer the best hope for bridging
the traditional gap that exists between experimental approach, the theoretical approach and
computational modeling for studying and understanding the behavior of materials.
Owing to the central role that multiple scale methodology appears poised to play in
the computational mechanics and materials science in the foreseeable future, this book
aims to summarize the past and the current developments in multiple scale modeling to
provide a coherent starting point from which interested scientists and engineers can begin
their journey into this vast and rapidly expanding subject. We hope that this book is one
of the first systematic works aimed at providing knowledge about fundamental concepts
behind nanoscale mechanics and materials and the relevant applications. The book contains
both published and previously unpublished material and is aimed at nanoscale engineers,
designers, materials scientists and interested students and researchers.
A salient feature of this book is that it is also intended to be used as an educational
tool. The major reason is to synthesize the state of the art in multiple scale modeling
techniques into the classroom such that the crucial tools being made available today are
passed onto the next generation of scientists and engineers. Thus, the materials in this book
which were previously used for courses at Northwestern University and the National Science
Foundation (NSF) Summer Institute on Nano Mechanics and Materials have been coherently
combined with Powerpoint lecture notes and selected computer codes (available online at
www.wiley.com/go/nanomechanics) to make the material presented readily accessible for
those researchers who are interested in joining and contributing to the field of multiple
scale modeling and analysis. Along with the review of basic theoretical concepts, they
present the solutions and dynamic visualization of numerous practical problems, ranging
from simple one-dimensional systems to state-of-the-art applications. The solutions of the
simple illustrative problems are augmented by Matlab and Mathematica codes which serve
to highlight the numerical implementation of the theoretical approaches presented in this
book.
There are many other novel and unique aspects to this book. As mentioned above,
the integration of teaching and research is one of the key features. The material contains
detailed expositions on all the topics that are necessary to fully comprehend multiple scale
analysis. As such, the book is logically divided into three parts. The first part consists
of Chapters 2–4, which cover the theoretical basis needed to understand the behavior of
multiparticle atomistic systems. The second part consists of Chapters 5–8, and introduces
multiple scale methods. In particular, the bridging scale concurrent approach, which is
based on the theoretical considerations provided in the first part of the book, is given
special attention here. The third part comprising Chapters 9–10 is devoted to contemporary
applications in the area of nanostructured and bioinspired materials, biofluidics and cell
mechanics.
Chapter 1 contains an introduction, and emphasizes the need for multiple scale simulations
by presenting case studies from different scientific disciplines, including materials
design and biofluidics. Chapter 2 introduces the notion of Lagrangian dynamics description
of systems of interacting particles, including nonconservative equations of motion, multibody
interatomic potentials and arbitrary molecular shapes. Chapter 3 details the extension
of the Lagrangian method to spatially periodic lattice structures; it reviews the relevant symmetry
concepts, and derives the basic response solutions for a general three-dimensional
lattice in semianalytical forms that are important in nanoscale engineering applications.
Chapter 4 gives a systematic, though condensed, exposition on contemporary approaches
that allow an averaged macroscopic characterization of multiparticle systems in thermodynamic
equilibrium; these include the methods of thermodynamic potentials, statistical
averaging, microcanonical and canonical ensemble theories.
Chapter 5 provides an overview of multiple scale modeling. As such, previously developed
multiple scale methods are reviewed and analyzed, and capabilities that are needed in
multiple scale modeling are discussed and provided as a basis for the remaining chapters.
Chapter 6 introduces the bridging scale concurrent method, which couples atomistic and
continuum scale models; here, connections are made between the bridging scale, particle
dynamics and lattice mechanics concepts introduced in Chapters 2–3.
Numerical validation of the bridging scale approach is given in Chapter 7. The numerical
examples in one, two and three dimensions highlight the applicability of the bridging scale
to highly nonlinear physical phenomena, including the fracture and subsequent failure of
materials. The recent extension of the bridging scale to incorporate quantum mechanical
information into the coupling of length scales framework is also described in this chapter.
Chapter 8 provides an extension of the MD impedance force such that it can be utilized
with long-ranged interatomic potentials; this extension is crucial as most realistic interatomic
potentials incorporate non-nearest neighbor bonding. This chapter concludes the section on
multiple scale modeling with comments on future research directions.
Chapter 9 highlights applications of multiple scale methods in crucial areas of physical
interest. In the realm of solids, the topics covered are the hierarchical and concurrent
design of realistic materials, including novel steel and metallic alloys, shape memory composites
and self-healing materials. Lastly, Chapter 10 emphasizes new research in the area
of computational biofluidics, electrohydrodynamics, bioengineering and nano-bio interfacial
problems. The topics include electrophoresis multiscale and multiphysics modeling of
red blood cell (RBC) aggregation and the effect on blood rheology, capillary flow, cell
migration, nanomanipulation and assembly of macromolecules.
We would like to thank our colleagues and graduate students for their contributions
to this book, in particular, Ted Belytschko, Antonio Bouze, Dmitry Dorofeev, David Farrell,
Mark Horstemeyer, Sukky Jun, Hiroshi Kadowaki, Adrian Kopacz, Yaling Liu, Cahal
McVeigh, Sergey Medyanik, Dong Qian, Leonid Shilkrot, Shaoqiang Tang, Franck Vernerey
and Sulin Zhang. Finally, we would like to thank and acknowledge the following sponsors
for their support: the National Science Foundation (NSF), the NSF Summer Institute
on Nano Mechanics and Materials, the NSF Integrative Graduate Education and Research
Traineeship (IGERT) program, the Army Research Office (ARO), the Office of Naval
Research (ONR) CyberSteel 2020 project and the ONR Nanofilament-Based Combined
Chemical/Biological Detectors Project.
Wing Kam Liu
Evanston, Illinois Eduard G. Karpov
Nashville, Tennessee Harold S. Park